Lp norm inequalities are established for multilinear integral
operators of Calderón-Zygmund type which incorporate oscillatory
factors eiP, where P is a real-valued polynomial. Our main
results concern nonsingular multilinear operators
Ll(f1,f2,...,fn) = òRmeilP(x)Õj=1nfj(pj(x))h(x) dx, where
lÎR, P is a measurable real-valued function, each
fjÎL¥, and hÎC10 is compactly supported. Each
pj denotes the orthogonal projection from Rm to a linear
subspace of Rm of arbitrary dimension k£m-1. Basic
questions concerning the asymptotics of such integrals as
|l|®¥ are posed but only partially answered. A related
problem concerning measures of sublevel sets is solved.

We study the spaces Qa(Rn), which are subpaces of BMO, and
their dyadic counterparts. We prove a quasi-orthogonal decomposition
for functions in Qa(Rn), analogous to that for functions in
BMO. This is joint work with Jie Xiao.

We consider the class of singular integral operators on the Heisenberg
group with radial convolution kernels satisfying standard
Calderón-Zygmund-type conditions. In particular, this class
includes the Cauchy-Szegö projection. Since their kernels are
radial, the operators in this class can also be described as spectral
multipliers. They form a sub-class of the Marcinkiewicz-type spectral
multipliers, whose convolution kernels were characterized by Müller,
Ricci, and Stein in 1995. I establish a condition on the multipliers
which characterizes this sub-class.

One-bit quantization refers to a class of algorithms widely used in
analog-to-digital conversion to approximate bandlimited functions by
local averages of {-1,+1} sequences on dense grids. The fair-duel
problem (which the speaker heard from S. Konyagin) asks for a universal
ordering of shootings that makes a duel between two equal and
bad-shooter duelists as fair as possible, with no prior probabilistic
information. In this talk, we present the links between these problems
and report some of the recent progress. We also discuss relations to
some other extremal problems on {-1,+1} sequences.

We shall give a survey of recent results establishing criteria for the
Lp boundedness of Riesz transforms associated to certain elliptic
operators (for example, divergence form operators on Rn, or the
Laplace-Beltrami operator on a complete non-compact Riemannian
manifold; we note, however, that the results under discussion are not
really about the structure of the particular operator or manifold, but
concern rather the relationship between estimates for the Riesz
transforms, and estimates for the associated heat kernels).

The starting point is an L2 estimate, which for the Laplace-Beltrami
operator is immediate from self-adjointness, and in the case of
divergence form operators is the recent solution of the Kato square
root problem. The Lp theory can therefore be viewed as the
development of some sort of Calderon-Zygmund machinery, in the absence
of standard regularity estimates for the singular kernels. The cases
p > 2 and p < 2 are essentially different, and the corresponding
theories have developed independently. Contributors to this subject
(sometimes jointly and sometimes independently, in various
combinations) include Auscher, Coulhon, Duong, Blunck, Kuntsmann and
Martell.

defined for smooth functions on the plane and measurable vector fields
v from the plane into the unit circle. We prove that if v has
1+e derivatives, then Hv extends to a bounded map from
L2(R2) into itself. The norm of Hv grows logarithmically
in the C1+e norm of v.

What is noteworthy is that this result holds in the absence of some
additional geometric condition imposed upon v, and that the
smoothness condition is nearly optimal.

Whereas Hv is a Radon transform, for which there is an extensive
theory, our methods of proof are necessarily those associated to
Carleson's theorem on Fourier series, and the proof given by Lacey and
Thiele. These ideas can be adapted to the study o Hv. We find it
necessary to combine them with a crucial maximal function estimate
that is particular to the smooth vector field in question.

We study the well-posedness of the Dirichlet and Neumann problems for
the Laplace-Beltrami operator in a Lipschitz sub-domain of a smooth,
compact manifold, equipped with a rough metric tensor. More
specifically, the aim is to derive sharp estimates on Sobolev and Besov
spaces when the metric tensor has a modulus of continuity satisfying a
Holder or a Dini-type condition. This is joint work with Michael
Taylor.

Hardy and Littlewood observed that Lp-spaces on the torus have the
majorant property if p is a positive even integer. For p > 2 not an
even integer it is known that the majorant property fails to hold. We
will discuss a linearized variant of the majorant problem which relates
it to local restriction problems for Fourier series with
frequency set in [0,N]. For a random selection of frequency
sets Ew in [0,N] of size Na, 0 < a < 1, we show that for
e > 0 the events

sup|an|£ 1

||

ånÎEw

ane2pinx||p£CeNe||

ånÎEw

e2pinx||p

have probability that tends to 1 as N®¥. However, for
certain frequency sets E in [0,N] we will show that above estimate
fails by a positive power in N.

This talk will be a survey recent results and unsolved problems in
additive number theory. The first part will consider sums of finite
sets of integers and lattice points, and, more generally, of finite
subsets of arbitrary abelian semigroups. Of particular interest is the
asymptotic geometric behavior of h-fold sumsets as h tends to
infinity.

The second part will consider sums of infinite sets of integers and
lattice points. We consider various extremal problems of h-fold
sumsets, as well as the classification of representation functions of
asymptotic bases of finite order for the integers and the nonnegative
integers.

In this work joint with Andreas Seeger, we consider the Lp
regularity of an averaging operator over a curve in R3 with
nonvanishing curvature and torsion. We also prove related local
smoothing estimates, which lead to Lp boundedness of a certain
maximal function associated to these averages. The common thread
underlying the proof of these results is a deep theorem of T. Wolff on
cone multipliers.

We will consider in this lecture the inverse boundary problem of
Electrical Impedance Tomography (EIT). This inverse method consists in
determining the electrical conductivity inside a body by making voltage
and current measurements at the boundary. The boundary information is
encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is
to determine the coefficients of the conductivity equation (an elliptic
partial differential equation) knowing the DN map. In particular we
will consider EIT for anisotropic conductivities (the conductivity
depends on direction), which can be formulated, in dimension three or
larger, as the question of determining a Riemannian metric from the
associated DN map. We will discuss a connection of this latter problem
with the boundary rigidity problem. In this case the information is
encoded in the boundary distance function which measures the lengths of
geodesics joining points in the boundary of a compact Riemannian
manifold with boundary.